I am trying to solve the following question:
Let $G$ be a group and $(A, +)$ be an abelian group. For $f,g \in Hom(G,A)$ and $x \in G$ define $(f + g)(x) = f(x) +_A g(x)$.
(a) Show that $f+g \in Hom(G,A)$.
(b) Show that $(Hom(G,A),+)$ is an abelian group.
(c) Let $G$ be a group, and let $id: \ G \rightarrow G$ be the identity homomorphism. Define $f: G \rightarrow G$ by $f(x) = (id(x))(id(x)) = x \cdot x = x^2$. Suppose that $f \in Hom(G,G)$. Show that $G$ is commutative.
In part (a), what does it even mean to say that one adds homomorphisms $f+g$? I understand the idea of adding two functions $f,g \in \mathbb{R}^{\mathbb{R}}$ for instance, since we can add polynomials, but to add homomorphisms makes no sense to me.
Let's make it clear;
We are not adding homomorphisms ;we are adding functions which satisfy some desired properties i.e $f(x+_Gy)=f(x)+_Af(y)$ where $+_G,+_A$ represent operations in $G,A$ respectively.
For the problem; $(f+g)(x+y)=f(x+y)+g(x+y)$[By definition of $f+g$]=$f(x)+f(y)+g(x)+g(y)=(f+g)(x)+(f+g)(y)$.